Gallai's path decomposition conjecture for graphs with treewidth at most 3

Abstract

A path decomposition of a graph G is a set of edge‐disjoint paths of G that covers the edge set of G. Gallai (1968) conjectured that every connected graph with n vertices admits a path decomposition of size at most ⌊(n+1)∕2⌋. Gallai's conjecture was verified for many classes of graphs. In particular, Lovász (1968) verified this conjecture for graphs with at most one vertex of even degree, and Pyber (1996) verified it for graphs in which every cycle contains a vertex of odd degree. Recently, Bonamy and Perrett verified Gallai's conjecture for graphs with maximum degree at most 5. In this paper, we verify Gallai's conjecture for graphs with treewidth at most 3. Moreover, we show that the only graphs with treewidth at most 3 that do not admit a path decomposition of size at most ⌊n∕2⌋ are isomorphic to K3 or K5−, the graph obtained from K5 by removing an edge.

Towards Gallai's path decomposition conjecture

“…In fact, knowing that the only non‐Gallai graphs with E‐degree at most 3 are the odd semi‐cliques would simplify the proof of Claim 3, which was introduced to deal with SET subgraphs. By using the Integer Linear Formulation presented in [8], we were able to check this fact for SET graphs up to eleven vertices.…”

“…Such graphs are known as odd semicliques [1]. This motivates the following strengthening of Conjecture 1, which was considered in [6,7].…”

“…Graphs for which are called Gallai graphs . Botler et al [7] verified Conjecture 4 for graphs with treewidth at most 3 by proving that partial 3‐trees are either Gallai graphs, or one of the two odd semicliques that are partial 3‐trees ( and ). They also prove [4] that a graph with maximum degree at most 4 is either a Gallai graph, or one of the three odd semicliques with maximum degree at most 4 (, , and ).…”

Gallai's path decomposition in planar graphs

“…This notion of reducible subgraph has been used in previous works [5,7,8,10,11]. We present here a formalism appropriate for our subgraphs.…”

“…In 1968, Gallai stated this simple but surprising conjecture [1]: every graph on n vertices admits a ⌈ n 2 ⌉-path decomposition. Gallai's conjecture is still unsolved as of today, and has only been confirmed on very specific graph classes: graphs in which each vertex has degree 2 or 4 [2], graphs whose vertices of even degree induce a forest [3], graphs for which each block of the subgraph induced by the vertices of even degree is triangle-free with maximum degree at most 3 [4,5], series-parallel graphs [6], or planar 3-trees [7]. Bonamy and Perrett confirmed the conjecture on graphs with maximum degree at most 5 [8], and Chu, Fan and Liu for graphs of maximum degree 6, under the condition that the vertices of degree 6 form an independent set [9].…”